Math Problem Statement

Correct?

What is the interpretation of the expression |2x - 3|, and how does it relate to the distance between points?

Solution

The steps shown in your image are essentially correct, but let's break it down and analyze the details.

You start with the expression: 2x3|2x - 3| This represents the absolute value, which can be interpreted as the distance between 2x2x and 33 on a number line.

Steps Explanation:

  1. Expression 2x3|2x - 3|: This is interpreted as the absolute distance between 2x2x and 33. So, the original expression is correct in describing the "distance between 2x2x and 3."

  2. Factoring out 1-1: The next step is factoring out the 1-1 from inside the absolute value expression: 1(2x+3)|-1(-2x + 3)| This is mathematically sound because pulling out a negative factor doesn’t change the overall value inside the absolute value. The absolute value function ignores signs.

  3. Simplification 2x+3|-2x + 3|: You remove the 1-1 because: 1(2x+3)=2x+3|-1 \cdot (-2x + 3)| = | -2x + 3 | So this is also correct. The absolute value of a negative or positive expression is the same as that of the original expression.

  4. Conclusion: The expression 2x+3|-2x + 3| represents the distance between 2x-2x and 3.

Final Verdict:

Your steps are correct, and the final result shows that 2x3|2x - 3| is equivalent to 2x+3|-2x + 3|, as both represent the same distance.

Do you want details on how absolute values represent distances or any clarifications? Here are five questions to help expand your understanding:

  1. What is the geometric interpretation of absolute value on a number line?
  2. How does factoring affect expressions inside an absolute value?
  3. Why is a=a|a| = |-a| true for any number aa?
  4. Can we generalize the property that f(x)=f(x)|f(x)| = |-f(x)| for any linear function f(x)f(x)?
  5. What happens if we add or subtract constants from absolute value expressions?

Tip: Remember, the absolute value of any real number is its distance from 0, regardless of direction on the number line!

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value
Factoring

Formulas

|a| = distance between a and 0
|a - b| = distance between a and b

Theorems

Absolute Value Property
Distance Formula

Suitable Grade Level

Grades 7-10